February 1997 Volume 24 Number 1
Lego of my simplex
By Norman PendegraftIn an article in OR/MS Today , Robert Bosch described an exercise for an introductory management science course in which he requires the students to create a low-cost nutritionally complete diet from McDonalds.
I have used this exercise and find it extremely effective in demonstrating some of the difficulties of solving real problems. Since many students find that concrete examples help them to understand abstract ideas, I have developed an in-class exercise using the popular children's toy, Legos, to illustrate the economics of linear programming during a one-hour lecture.
The class divides into small groups, and each group is given a bag containing eight small Legos and six large Legos. I demonstrate the construction of a table and a chair (see Figure 1).
Prices for tables and chairs are announced ($16 and $10 in the current example), and the class is asked to select a product mix to maximize their profits using the available resources. After a few minutes, most teams have found the optimal mix &endash; two tables and two chairs (see Figure 2). Some have stopped with a suboptimal solution of three tables.
Figure 2: The Initial Solution
The simplex method involves a three-step search procedure: find the activity with the highest marginal value, find the tightest constraint, substitute one activity for another. If we start with a solution of zero chairs and zero tables (see Figure 3) and then employ this procedure, we will find a solution of three tables with a profit of $48. The second simplex table shows that chairs now have a marginal value of $2 rather than their initial $10. Why should this be so?
Figure 3: The Optimal Solution
Explaining why the marginal values of the activities change and the mechanics of the substitution is always hard at the blackboard, but when the students have three tables in hand, they can easily see that in order to make any chairs, they must first take apart a table in order to get the needed resources. To make a chair, they must give up half a table, which gives chairs a marginal value of $10-(1/2)=$2.
When they reach the next extreme point, they can see that while they can make more chairs, the exchange rate changes; because they are now out of small Legos, they must now give up one table for each chair. This means that the marginal values change, or that another pivot is needed.
They also see why this point is optimal: the value of the table given up ($16) exceeds the value of the new chair ($10). Now that we have an optimal solution, we can explore the sensitivity of this solution to changes in the amounts of available resources. I hold aloft another large Lego and ask what I am bid for the extra resource. Some will immediately bid $16 because they recognize that they can use it to make a new table. Then they realize that they must give up a chair to do so, and find the correct marginal value, $6.
When I ask how many they would buy at that price they can quickly see by disassembling their own tables and chairs that there is no value to more than two large ones; i.e. the allowable increase is two. Similarly, they find that the allowable decrease is two. Later when we work through this calculation, we refer back to the class exercise. Student response to the exercise has been extremely favorable.
1. Bosch, Robert A., 1993, "Big Mac Attack," OR/MS Today, August, pp.30-31.
Norman Pendegraft is a professor at the College of Business and Economics at the University of Idaho , Moscow, Idaho.
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